On the Killing vector fields of generalized metrics
Rezs˝o L. Lovas
(Received September 9, 2004)
Abstract We consider a manifold endowed with a metric tensor in its tangent
bundle pulled back by its own projection. We shall give necessary and sufficient conditions for a vector field to be an infinitesimal isometry of a metric of this type in general and for some special classes. We also examine translations, i.e., the special class of Killing vector fields whose integral curves are geodesics of an associated Finsler manifold. As applications, we determine the Killing vector fields of Funk metrics, and we give a new proof for the fact that perturbing a Riemannian manifold by a one-form metrically equivalent to a Killing field yields a Randers manifold for which the original vector field is a Killing field as well.
AMS 2000 Mathematics Subject Classification. 53B40.
Key words and phrases. Generalized metrics, Killing vector fields, translations, Randers manifolds, Funk metrics.
§1. Introduction
By a generalized metric we shall mean a symmetric, non-degenerate (0, 2) tensor in the pull-back bundle τ∗τ of the tangent bundle τ : T M → M over τ . The study of metrics of this type dates back to the 1950’s [13, 25]. A new classification for them has been published recently [10]. These metrics are natural generalizations of Finsler structures, since manifolds endowed with generalized metrics are the most general spaces where ‘the metric depends also on the direction’. Some of their characteristic properties in which they differ from Finsler manifolds were already pointed out in [13], e.g., the fact that their autoparallel and extremal curves do not necessarily coincide, even with a natural choice of a covariant derivative. These metrics may be interesting not only from a geometrical, but also from a physical viewpoint, since they furnish a natural geometric description of the so-called bilocal field theories introduced by Yukawa in the 1940’s. Yukawa’s main goal was to explain mass
quantization and to eliminate certain types of divergences in quantum field theory. For bilocal field theories, we may refer to Yukawa’s original papers [27, 28], or, for more recent reviews on multi-local theories, see [15, 22]. In this paper, however, we restrict ourselves to the geometric aspects of generalized metrics; we wish to consider physical implications in a later article.
The infinitesimal symmetries of space-time are expressed by so-called Kil-ling vector fields in general relativity. Therefore, it is an important problem to determine the Killing vector fields of different classes of generalized metrics. In a Euclidean space, translations are distinguished from other types of isometries by the property that their orbits are straight lines. This property is used to generalize the notion of translations to more general classes of metrics: translations are Killing vector fields whose integral curves are at the same time geodesics (in some sense). In this paper we also study the translations of a certain type of generalized metrics.
The outline of the paper is the following. Sections 2 – 4 may be regarded as preparatory sections, since they contain no new results; they only make the paper more or less self-contained. Coming to the original results, in section 5 we have collected those which are relevant to all generalized metrics. We discuss the Killing vector fields of special types of metrics in section 6. In section 7 we study the translations of weakly normal and Miron regular met-rics. Section 8 contains applications to Randers manifolds and Funk metmet-rics. Finally, in section 9 we discuss some open problems.
§2. Preliminary constructions
We begin by recalling some definitions and basic facts concerning the technical tools that we shall use later. As a general reference, see [8, 21].
We work on an n-dimensional connected smooth manifold M whose topol-ogy is of Hausdorff type and has a countable base. The symbol C∞(M ) stands for the ring of smooth real-valued functions on M , andX(M) is the C∞(M )-module of (smooth) vector fields on M . The symbol τ : T M → M is the tangent bundle of M , and the tangent bundle of T M is denoted by τT M. We
shall denote the open submanifold of T M formed by the non-zero tangent vectors by T M , and the restriction of τ to◦ T M by◦ τ . If N is another mani-◦ fold, and f : M → N is a smooth map, then its tangent map is denoted by f∗ : T M → T N. If f is a diffeomorphism, the push-forward of a vector field X on M by f is
fX := f∗◦ X ◦ f−1.
A subset W of the product manifoldR × M is said to be radial if, for any p∈ M, W ∩(R×{p}) = I×{p}, where I is an open interval that contains 0 ∈ R.
Let X be a vector field on a manifold M . The flow of X is a map ϕ : W → M such that W ⊂ R × M is a radial set, and cp := ϕ(., p) : Ip → M is the
maximal integral curve of X starting from the point p∈ M, i.e., ˙cp = X◦ cp,
cp(0) = p, and any other curve satisfying these two conditions is a restriction of cp. If W =R × M, the vector field X is said to be complete.
If f is a smooth function on M , then the function fc : T M → R, v∈ T M → fc(v) := vf
is a smooth function on T M and is called the complete lift of f . It can be shown that any vector field on T M is determined by its action on complete lifts, and if X ∈ X(M), there is a unique vector field Xc on T M such that Xcfc = (Xf )c for any smooth function f on M [21]. The vector field Xc is said to be the complete lift of X. Let ϕ : W → M be the flow of X. If we fix the first argument of ϕ, the map ϕt:= ϕ(t, .) is a diffeomorphism between two open submanifolds of M , and the map
˜
ϕ : (t, v)→ ˜ϕ(t, v) := (ϕt)∗(v) ((t, τ (v))∈ W )
is the flow of Xc.
The pull-back bundles of τ by τ and τ will play an important role in our◦ presentation, and will be denoted by τ∗τ andτ◦∗τ , respectively. The shorthand for their sections will be X(τ) and X(τ ). These sections will also be called◦ vector fields along the projection.
We have the canonical short exact sequence
0→ τ∗T M → T T Mi → τj ∗T M → 0,
where i(z, v) is the initial velocity of the parametrized straight line t→ z + tv for all (z, v) ∈ τ∗T M , and j is defined by w ∈ TzT M → (z, τ∗(w)). The set of vertical vectors is V T M := Im i = Ker j, it is the total space of the vertical subbundle of τT M, denoted by τT Mv . The module of the vertical vector fields
is Xv(T M ). Note that the Lie bracket of two vertical vector fields is always vertical.
The bundle maps i and j give rise to C∞(T M )-homomorphisms between X(τ) and X(T M) denoted by the same symbols. Thus we obtain the exact sequence
0→ X(τ)→ X(T M)i → X(τ) → 0j of C∞(T M )-homomorphisms.
If X is a vector field on M , we define ˆ
Obviously, ˆX is a vector field along τ , while Xv is a vertical vector field. The vector field ˆX is said to be a basic vector field along τ , and Xv is called the vertical lift of X. Further important canonical objects are given by
δ(z) := (z, z) (z∈ T M), C := iδ, and J := i◦ j,
the canonical section of τ∗τ , the Liouville vector field on T M and the vertical endomorphism, respectively. We associate to J the vertical differential dJ on T M . By definition,
dJf := df ◦ J, f ∈ C∞(T M ).
Then dJf is a (semibasic) one-form on T M .
If α ∈ T0k(M ) is a symmetric or skew-symmetric k-form on M , then the tensor fields ˆα and ¯α defined by
ˆ
αv(v1, . . . , vk) := αp(v1, . . . , vk), α¯v(v1, . . . , vk−1) := αp(v, v1, . . . , vk−1) (v, vi ∈ TpM, 1 i k; p ∈ M)
are symmetric or skew-symmetric k- and (k− 1)-forms along τ, respectively. In particular, if f ∈ C∞(M ), then fv := ˆf = f◦ τ ∈ C∞(T M ) is the vertical lift of f .
Let ˜X and ˜Y be two vector fields along τ . Choose a vector field η on T M such that jη = ˜Y . We define the canonical v-covariant derivative of ˜Y with respect to ˜X by ∇v ˜ XY =˜ ∇vX˜jη := j i ˜X, η .
It can easily be seen that the definition is independent of the choice of η. The operator∇vX˜ can be extended to any tensor α of type (0, s) along τ , to be a kind of tensor derivation:
(∇vX˜α) ˜ Y1, . . . , ˜Ys := i ˜X α ˜ Y1, . . . , ˜Ys − s i=1 α ˜ Y1, . . . ,∇vX˜Y˜i, . . . , ˜Ys ˜ Y1, . . . , ˜Ys∈ X(τ) .
If X is a vector field on M , we may define a Lie derivativeLX in the tensor
algebra of τ∗τ in the following way:
LX : f ∈ C∞(T M )→ Xcf, Y˜ ∈ X(τ) → i−1
Xc, i ˜Y
,
and extend it to any types of tensors by the usual product rule (for details, see [6, 21]). In particular, LXδ = 0, and, for Y ∈ X(M), we have
§3. Generalized metrics
In this section we introduce generalized metrics and some of their special classes. Our main source is reference [10].
Definition 3.1. Let g be a symmetric and non-degenerate tensor of type (0, 2) in the bundle τ∗τ or inτ◦∗τ . Then g is said to be a generalized metric or briefly a metric.
It is crucial that g need not be defined on the zero section, since, if g is homogeneous and is defined in the whole τ∗τ (and, of course, is smooth), then it is the lift of a pseudo-Riemannian metric on M .
Using non-degeneracy, the first Cartan tensor C and the lowered first Car-tan tensor C of a generalized metric g are defined by the following formulae:
g C( ˜X, ˜Y ), ˜Z :=C ˜ X, ˜Y , ˜Z := ∇v ˜ Xg ˜ Y , ˜Z ˜ X, ˜Y , ˜Z ∈ X(τ) . The one-form ϑg : ξ∈ X(T M) → ϑg(ξ) := g(jξ, δ)
on T M is called the Lagrange one-form associated to g, and its exterior deriva-tive ωg := dϑg is the Lagrange two-form associated to g. The absolute energy of g is E := 12g(δ, δ).
Definition 3.2. A metric g along τ or τ is said to be variational if the first◦ Cartan tensor C associated to it is symmetric, weakly variational if
C ˜ X, ˜Y , δ =C ˜ Y , ˜X, δ
for every ˜X, ˜Y ∈ X(τ), normal if C
˜ X, δ
= 0 for every ˜X ∈ X(τ), and weakly normal if C
˜ X, δ, δ
= 0 for every ˜X ∈ X(τ). The metric is Miron regular [12] if the tensor
˜ B : ˜X∈ X(τ) → ˜B ˜ X := ˜X +C ˜ X, δ
has maximal rank at every point of T M (or T M ).◦
Now, for the sake of the reader’s convenience, we summarize some results of [10] we shall make use of.
(1) A metric g is variational if and only if there is a smooth function L on T M (or onT M ) such that g =◦ ∇v∇vL. In this case, we shall call L a Lagrangian.
(2) A metric g is weakly variational if and only if there is a smooth function L on T M (or onT M ) such that ϑ◦ g= dJL.
(3) If g is weakly normal and Miron regular, then E is positively homoge-neous of degree 2, and the symmetric tensor∇v∇vE is non-degenerate. In other words, E is a (possibly indefinite) Finsler energy function. Fur-thermore, ϑg = dJE.
(4) If g is normal, then there is a (possibly indefinite) Finsler energy function E such that g =∇v∇vE.
§4. Ehresmann connections and covariant derivatives
Following the terminology used e.g. in [7], by an Ehresmann connection we shall mean a split canonical short exact sequence:
0 τ∗T M i V T T M j H τ ∗T M 0.
The requirement that this is a splitting means that V ◦ i = j ◦ H = 1τ∗T M, and ImH = Ker V. We allow the possibility that H and V are defined only on
◦
T M rather than on the whole T M . The type (1, 1) tensor field h :=H ◦ j on T M is said to be the horizontal projector belonging toH, and Im hv is called
the horizontal subspace of TvT M if v ∈ T M. The map v := 1T M − h is the
vertical projector belonging to h. As in the case of i and j, we denote by the same symbols the arising C∞(T M )-homomorphism between the modules of vector fields as the corresponding bundle maps. If X∈ X(M) is a vector field on M , then Xh :=H ˆX = hXc ∈ X(T M) is its horizontal lift.
The torsion of an Ehresmann connection is the (1,2) tensor T along τ determined by the formula
iT ˆ X, ˆY := Xh, Yv −Yh, Xv − [X, Y ]v (X, Y ∈ X(M)). If a metric and an Ehresmann connection with vanishing torsion are given on T M (or onT M ), we can construct a metric covariant derivative D in τ◦ ∗τ as follows (see [4, 10]). First, we consider Berwald’s covariant derivative in τ∗τ given by ∇i ˜XY := j˜ i ˜X,H ˜Y, ∇ H ˜XY :=˜ V H ˜X, i ˜Y ˜ X, ˜Y ∈ X(τ )◦ . Observe that its vertical part coincides with the canonical v-covariant deriva-tive. Next, we introduce the second Cartan tensor Ch by means of the relation
g ChX, ˜˜ Y, ˜Z:= (∇ H ˜Xg) ˜ Y , ˜Z ˜ X, ˜Y , ˜Z ∈ X(τ) .
Third, using the Christoffel trick, we define two other tensors along τ : g ◦ C( ˜X, ˜Y ), ˜Z = g C( ˜X, ˜Y ), ˜Z + g C( ˜Y , ˜Z), ˜X − gC( ˜Z, ˜X), ˜Y , g ◦ Ch( ˜X, ˜Y ), ˜Z= gCh( ˜X, ˜Y ), ˜Z+ gCh( ˜Y , ˜Z), ˜X− gCh( ˜Z, ˜X), ˜Y.
With the help ofC and◦ C◦h we define D by the rules Di ˜XY :=˜ ∇i ˜XY +˜ 1 2 ◦ CX, ˜˜ Y, D H ˜XY :=˜ ∇H ˜XY +˜ 1 2 ◦ ChX, ˜˜ Y ˜ X, ˜Y , ˜Z ∈ X(τ )◦ .
Finally, this covariant derivative operator can also be extended to any type of tensors by the usual product rule. Then it will be metric, i.e., Dg = 0. If g arises from a Finsler energy function, and H is the canonical Ehresmann connection on the Finsler manifold (section 7), then D coincides with the well-known Cartan’s covariant derivative [20, 21].
§5. Killing vector fields in general
In this section g will be a generalized metric on M . For the sake of definiteness, we shall assume that g is defined only onT M . The same arguments, however,◦ remain valid when its domain is the whole T M .
Definition 5.1. A diffeomorphism f : U → V between two open subsets of M is a local isometry if its tangent map leaves g invariant, i.e.,
gf∗(v)(f∗(w1), f∗(w2)) = gv(w1, w2)
for any p ∈ U and v, w1, w2 ∈ T◦pM . A vector field X ∈ X(M) with flow ϕ : W ⊂ R × M → M is said to be an infinitesimal isometry if ϕt is a local
isometry between two open subsets of M for all t∈ R such that the domain of ϕt is not empty. A vector field X ∈ X(M) is called a Killing vector field if LXg = 0.
Proposition 5.2. Let g be a metric and X ∈ X(M) a vector field. Then X is an infinitesimal isometry of M if and only if it is a Killing vector field. Proof. We shall repeatedly use the dynamic interpretation of the Lie bracket of two vector fields [24]: if X, Y ∈ X(M), and ϕ is the flow of X, then
[X, Y ](p) = lim t→0 1 t{(ϕ−t)∗[Y (ϕt(p))]− Y (p)} = limt→0 1 t((ϕ−t)Y − Y )(p),
for all p∈ M. Now let us begin with proving the necessity, and assume that X is an infinitesimal isometry. For arbitrarily chosen vector fields Y and Z on M , define a function f ∈ C∞T M◦ by f := g ˆ Y , ˆZ . If v∈T◦pM , t∈ R and (t, p)∈ W , we have f ((ϕt)∗v) = g(ϕt)∗(v)(Y (ϕt(p)), Z(ϕt(p))) = g(ϕt)∗(v){(ϕt)∗[(ϕ−t)Y ](p), (ϕt)∗[(ϕ−t)Z](p)} = gv((ϕ−t)Y (p), (ϕ−t)Z(p)),
using, in the last step, that ϕt is a local isometry for every sufficiently small t ∈ R. Now we use the fact that the curve cv : t → (ϕt)∗(v) is an integral
curve of Xc to obtain Xc(v)f = lim t→0 1 t[f ((ϕt)∗(v))− f(v)] = lim t→0 1 t[gv((ϕ−t)Y (p), (ϕ−t)Z(p))− gv(Y (p), Z(p))] = lim t→0 gv((ϕ−t)Y (p)− Y (p), (ϕ−t)Z(p)) t + gv(Y (p), (ϕ−t)Z(p)− Z(p)) t = gv lim t→0 1 t((ϕ−t)Y (p)− Y (p)), limt→0(ϕ−t)Z(p) + gv Y (p), lim t→0 1 t((ϕ−t)Z(p)− Z(p)) = gv([X, Y ](p), Z(p)) + gv(Y (p), [X, Z](p)) = g [X, Y ], ˆZ + g ˆ Y , [X, Z] (v), Xcg ˆ Y , ˆZ = Xcf = g [X, Y ], ˆZ + g ˆ Y , [X, Z] = g LXY , ˆˆ Z + g ˆ Y ,LXZˆ . Thus we conclude (LXg) ˆ Y , ˆZ = Xcg ˆ Y , ˆZ − gLXY , ˆˆ Z − gY ,ˆ LXZˆ= 0, i.e., X is a Killing vector field.
To prove the converse, assume that X is a Killing vector field, consider the flow ϕ : W ⊂ R×M → M of X, and let p ∈ M, v, w1, w2 ∈T◦pM be arbitrary.
We shall again denote the maximal integral curve of Xc starting from v by cv : Ip → T M. (The domain of this curve depends only on p.) We define the
function : Ip→ R in the following way:
It is enough to show that is constant. To this end, we define two vector fields along cv:
Y (t) := (ϕt)∗(w1), Z(t) := (ϕt)∗(w2) (t∈ Ip).
Then Y and Z can be extended, at least locally, to vector fields ˜Y and ˜Z on an open subset U of T M such that
Y (t) = ˜Y (cv(t)), Z(t) = ˜Z(cv(t)) (t∈ I)
(I⊂ Ip is another open interval). Now with the help of the function given by f (q) := gq( ˜Y (q), ˜Z(q)) (q∈ U), we have I = f ◦ cv. Thus, (t) = (f◦ cv)(t) = ˙cv(t)f = Xc(cv(t))f = (Xcf )(cv(t)) = g(LXY , ˜˜ Z) + g( ˜Y ,LXZ)˜ (cv(t)), i LXY˜ (q) = Xc, i ˜Y (q) = lim t→0 1 t (ϕ−t)∗[i ˜Y (ϕt(q))]− i ˜Y (q) = i lim t→0 1 t (ϕ−t)∗[ ˜Y (ϕt(q))]− ˜Y (q) = 0 (q∈ cv(I)) due to the construction of ˜Y . We obtain, in a similar way, that LXZ = 0.˜
Hence is indeed constant.
If the metric g is positive definite and homogeneous, i.e., the function g
ˆ X, ˆY
is positively homogeneous of degree 0 for any X, Y ∈ X(M), then we may define the length of an arc c : [α, β]→ M by
(c) := β α √ E◦ ˙c = β α g˙c(t)( ˙c(t), ˙c(t))dt. The distance of two points p, q∈ M is then given by
d(p, q) := inf{ (c)|c : [0, 1] → M, c(0) = p, c(1) = q}.
We say that g is reversible if g−v(w1, w2) = gv(w1, w2) for any v, w1, w2 ∈ TpM and p∈ M. In this case, d is symmetric, and (M, d) becomes a metric space. It is known that every Killing field is complete on a complete Riemannian manifold [17]. This result can be easily generalized as follows.
Proposition 5.3. Let g be a homogeneous, reversible and positive definite metric, and suppose that X is a Killing vector field of g. If M is complete as a metric space, the vector field X is complete as well.
Proof. Let cp : [0, α[→ M be an integral curve of X starting from p. We show that cp can be extended to [0, α]. Since ¨cp = Xc◦ ˙cp, and
XcE = 1 2X
cg(δ, δ) = 1
2(LXg)(δ, δ) = 0,
the function E◦ ˙cp is constant. Let λ := E( ˙cp(t)) (t ∈ [0, α[ is arbitrary).
Thus, if t, t ∈ ]0, α[, d(cp(t), cp(t)) t t E◦ ˙cp = λ|t − t|.
This implies, by the completeness of M , that the limit limt→αcp(t) exists.
Now we suppose that an Ehresmann connection is specified on M whose torsion vanishes. Let D be the covariant derivative operator constructed in section 4.
The following proposition was formulated in [19] for the special case of Finsler manifolds. It generalizes the skew-symmetry of the covariant differen-tial of a Killing field in Riemannian geometry.
Proposition 5.4. If X is a Killing vector field on M ,
g DH ˜YX, ˜ˆ Z + g ˜ Y , DH ˜ZXˆ + g CVXc, ˜Y , ˜Z = 0 for any ˜Y , ˜Z ∈ X(τ ).◦
Proof. Since the left-hand side is tensorial in ˜Y , ˜Z, it is enough to verify the formula for basic vector fields ˆY , ˆZ. Using the condition that X is a Killing
field, we obtain 0 = (LXg) ˆ Y , ˆZ = Xcg ˆ Y , ˆZ − g[X, Y ], ˆ Z − gY , ˆ [X, Z] Dg=0 = g DXcY , ˆˆ Z − g[X, Y ], ˆ Z + g ˆ Y , DXcZˆ − gY , ˆ [X, Z] = g ∇XhY +ˆ 1 2 ◦ ChX, ˆˆ Y+1 2 ◦ CVXc, ˆY − [X, Y ], ˆZ + (Y ↔ Z) = g VXh, Yv − [X, Y ]v+1 2 ◦ ChX, ˆˆ Y+1 2 ◦ CVXc, ˆY , ˆZ + (Y ↔ Z) T =0= gVYh, Xv+1 2 ◦ ChX, ˆˆ Y+1 2 ◦ CVXc, ˆY , ˆZ + (Y ↔ Z) = g ∇YhX +ˆ 1 2 ◦ ChY , ˆˆ X, ˆZ + 1 2 C VXc, ˆY , ˆZ+C ˆ Y , ˆZ,VXc − C ˆ Z,VXc, ˆY + (Y ↔ Z) = g DYhX, ˆˆ Z + g ˆ Y , DZhXˆ +C VXc, ˆY , ˆZ = g DYhX, ˆˆ Z + g ˆ Y , DZhXˆ + g CVXc, ˆY , ˆZ ,
where the symbol (Y ↔ Z) means an expression consisting of all preceding terms, with Y and Z interchanged.
§6. Special classes of generalized metrics
For any metric g, we introduce the (1,1) tensorC along τ by the prescription∗
∗ C : ˜X ∈ X(τ) → C ˜ X, δ , whereC is the first Cartan tensor of the metric.
Proposition 6.1. Let g be a weakly variatonal and Miron regular metric with ϑg = dJL. A vector field X on M is a Killing vector field for g if and only if the function XcL is a vertical lift and LX
∗
C = 0. Proof.
Suppose that X is a Killing field. If Y ∈ X(M), we have YvXcL = XcYvL− [Xc, Yv]L = Xc(dJL)(Yc)− dJL[X, Y ]c = Xcϑg(Yc)− ϑg[X, Y ]c = Xcg ˆ Y , δ − g[X, Y ], δ = (LXg) ˆ Y , δ = 0,
thus XcL is a vertical lift. To verify the necessity of the second condition, let Z be another vector field on M . Using our assumption LXg = 0
repeatedly, we get g (LX ∗ C)( ˆY ), ˆZ = g LX(C( ˆ∗ Y ))−C ∗[X, Y ], ˆZ = Xcg ∗ C( ˆY ), ˆZ − g ∗ C [X, Y ], ˆZ − g ∗ C( ˆY ), [X, Z] = Xcg C( ˆY , δ), ˆZ − gC( [X, Y ], δ), ˆZ − gC( ˆY , δ), [X, Z] = Xc ∇v ˆ Yg δ, ˆZ −∇v [X,Y ]g δ, ˆZ −∇v ˆ Yg δ, [X, Z] = XcYvg δ, ˆZ − XcgY , ˆˆ Z− [Xc, Yv]gδ, ˆZ+ g[X, Y ], ˆ Z − Yvg δ, [X, Z] + g ˆ Y , [X, Z] = YvXcg δ, ˆZ − Yvgδ, [X, Z]= Yv(L Xg) δ, ˆZ = 0,
which implies, by the non-degeneracy of g, thatLX ∗
C = 0. (2) Sufficiency
If XcL is a vertical lift, we obtain
(LXcϑg)(Yc) = XcYvL− [Xc, Yv]L = YvXcL = 0
for any vector field Y on M , which implies LXcϑg = 0. Since the Lie
derivative and the exterior derivative commute, we also have LXcωg =
LXcdϑg = 0. The second condition implies LXB =˜ LX
1 (τ)+C∗
= 0.
As LXg is tensorial, and g is Miron regular, it is sufficient to show that (LXg)
˜ B( ˆY ), ˆZ
ωg(J ξ, η) = g ˜ B(jξ), jη (ξ, η∈ X(T M)), we get (LXg) ˜ B( ˆY ), ˆZ = Xcg ˜ B( ˆY ), ˆZ − gLXB( ˆ˜ Y ), ˆZ − gB( ˆ˜ Y ), [X, Z] = Xcg ˜ B( ˆY ), ˆZ − gB ˜[X, Y ], ˆZ− gB( ˆ˜ Y ), [X, Z] = Xcωg(Yv, Zc)− ωg([Xc, Yv], Zc)− ωg(Yv, [Xc, Zc]) = (LXcωg)(Yv, Zc) = 0, thus concluding the proof.
The metric g does not determine L uniquely, since a vertical lift can be added to L without changing dJL. Moreover, we have
Corollary 6.2. With conditions similar to those in 6.1, if g is defined on the whole T M , and X is a Killing vector field, L can be chosen such that XcL = 0. Proof. By 6.1, there is a smooth function L on T M such that XcL is a vertical lift. Let us define L by
L(v) := L(v)− L(0τ(v)),
then L differs from L only by a vertical lift, and XcL = 0.
Now we introduce two canonical inclusions. The first one will be i1 : M → T M, p∈ M → i1(p) := 0p.
In other words, i1 is an embedding of M into T M that assigns to each point p the zero vector at p. The second inclusion is given by the prescription
i2 : T M → T T M, v∈ T M → i2(v) := ˙cv(0), where cv : t∈ R → 0τ(v) + tv.
We shall also use the shorthand ¯τ := i1◦ τ.
Proposition 6.3. Let g be a variational metric defined on the whole T M . A vector field X on M is a Killing vector field if and only if there is a Lagrangian L for g such that XcL = 0.
(1) Necessity
Suppose that X is a Killing vector field, and L is an arbitrary Lagrangian for g. Then we obtain
0 = (LXg) ˆ Y , ˆZ = Xcg ˆ Y , ˆZ − g[X, Y ], ˆ Z − gY , ˆ [X, Z] = XcYvZvL − [Xc, Yv]ZvL − Yv[Xc, Zv]L = YvZvXcL
for any vector fields Y, Z ∈ X(M). It follows that XcL is an affine function on each fibre. Now we define a new Lagrangian L by
L := L− L ◦ i1◦ τ − dL ◦ i2.
It is easy to see that the difference of L and L is also a fibrewise affine function, thus their Hessians are the same, i.e., g. We com-pute the action of Xc on the difference L− L over an induced chart (τ−1(U ), (xi)ni=1, (yi)ni=1) in T M by a chart (U, (ui)ni=1) in M :
Xc L ◦ i1◦ τ + dL ◦ i2 = X L ◦ i1 v +Xc dL◦ i2 = ⎡ ⎣Xi∂ L ◦ i1 ∂ui ⎤ ⎦ v +(Xi)v ∂ ∂xi dL◦ i2 +yj ∂Xi ∂uj v ∂ ∂yi dL◦ i2 = (Xi)v ∂ L ∂xi ◦ ¯τ +(Xi)vyj ∂2L ∂xi∂yj ◦ ¯τ +yj ∂Xi ∂uj v ∂ L ∂yi ◦ ¯τ .
This is a fibrewise affine function, just like XcL. To show that they are equal, it is enough to check that they coincide on the zero section and so do their linear parts on each fibre. The expression of XcL over our induced chart is XcL = (X˜ i)v∂ L ∂xi + y j∂Xi ∂uj v ∂ L ∂yi. Thus, XcL = XcL − Xc
L − Lvanishes indeed on the zero section: XcL ◦ i1− Xc L− L ◦ i1 = Xi ∂ L ∂xi ◦ i1 − Xi ∂ L ∂xi ◦ i1 = 0, whereas the linear part of XcL is
yi ∂ ∂yiX cL◦ ¯τ−(Xi)vyj ∂2L ∂xi∂yj ◦ ¯τ −yj∂Xi ∂uj v ∂ L ∂yi ◦ ¯τ = 0.
(2) Sufficiency (LXg) ˆ Y , ˆZ = Xcg ˆ Y , ˆZ − g[X, Y ], ˆ Z − gY , ˆ [X, Z] = XcYvZvL− [Xc, Yv]ZvL− Yv[Xc, Zv]L = YvZvXcL = 0.
Corollary 6.4. If (M, E) is a Finsler manifold with Finslerian metric g = ∇v∇vE, then a vector field X on M is a Killing vector field of g if and only
if XcE = 0.
§7. Translations
In this section we shall work on a manifold endowed with a weakly normal and Miron regular metric. It can be shown (see [10]) that in this case, the absolute energy E is a Finsler energy function. Then E can be extended continuously to the zero section. We shall denote by ξ∈ XT M◦ the canonical spray of the Finsler manifold (M, E) determined by the relation (ddJE)(ξ, η) =−ηE for
η∈ XT M◦ . It is well-known that there is a canonical Ehresmann connection on a Finsler manifold called the Barthel connection [21]. In this section we shall use this connection and the corresponding metric covariant derivative D. Then, for any vector field X on M ,
Xh = 1 2(X
c+ [Xv, ξ]), XhE = 0,
and ξ =Hδ is horizontal.
Definition 7.1. A Killing vector field X of g is called a translation if every non-constant integral curve of X is a geodesic of the Finsler manifold (M, E). For classical results on translations of Riemannian manifolds, see [2, 16, 26]. Now we generalize the important conservation lemma from Riemannian geometry ([14], p. 252) as follows.
Proposition 7.2. If X ∈ X(M) is a Killing vector field, and c : I → M is a geodesic of E, then the function
t∈ I → g˙c(t)(X(c(t)), ˙c(t)) is constant.
Proof. Let us denote the function in question by f . The curve ˙c is an integral curve of ξ, thus we have
f = ξg ˆ X, δ ◦ ˙c.
Using (3) in section 3 and the relation XcE = 0, we obtain ξg ˆ X, δ = ξϑg(Xc) = ξ(dJE)(Xc) = ξXvE =−XcE− XvξE + ξXvE =−2XhE = 0,
and therefore f = 0, which implies that f is constant.
Proposition 7.3. Let X be a Killing vector field of g. Then X is a translation if and only if the function
p∈ M → E(Xp) is constant.
Proof.
(1) Necessity
Suppose that X is a translation. If X = 0, the statement is obvious. Hence we assume that there is a point q ∈ M such that Xq = 0. We
define the following subset of M :
V :={p ∈ M|E(Xp) = E(Xq)}.
We shall show that V = M . First, V = ∅, since q ∈ V . Furthermore, V is closed, since it is the inverse image of the closed set{E(Xq)} ⊂ R
under the function
f : p∈ M → f(p) := E(Xp).
Thus it remains only to show that V is open.
To see this, take a point p∈ V . By the straightening-out theorem (see e.g. [1]), there is a chart (U, (ui)ni=1) around p such that X U = ∂u∂1. Consider an integral curve c : I → M of X, which is, by the definition of translations, a geodesic as well. Its components ci := ui◦ c have the following form:
c1(t) = c1(0) + t, ci(t) = ci(0) (2 i n).
On the other hand, c satisfies the differential equations of the geodesics: ci+ 2Gi◦ ˙c = 0,
where Gi= 1 2g ijyk ∂2E ∂xk∂yj − ∂E ∂xj ,
and (gij) is the inverse matrix of (gij). Putting these together, we infer
that Gi◦ ∂u∂1 = 0 on U . Since the matrix (gij) is non-degenerate, this implies that 0 = yk ∂ 2E ∂xk∂yj − ∂E ∂xj ◦ ∂ ∂u1 = ∂2E ∂x1∂yj − ∂E ∂xj ◦ ∂ ∂u1 =−∂E ∂xj ◦ ∂ ∂u1 =− ∂ ∂uj E◦ ∂ ∂u1 ,
which, in turn, implies that the function E◦∂u∂1 is constant on U . Hence p ∈ V is contained together with an open neighbourhood in V . We conclude that V = M .
(2) Sufficiency
If the function f : p∈ M → f(p) := E(Xp) is constant, then, in a chart
similar to that in the previous part, it can be seen that the integral curves of X are geodesics as well.
§8. Some special cases
8.1. Randers manifolds
Let (M, α) be a Riemannian manifold and β a one-form on M . We recall from section 2 that the tensor ˆα along τ and the function ¯β on T M are given by
ˆ
αv(w1, w2) = αp(w1, w2), β(v) = β¯ p(v) (v, w1, w2∈ TpM, p∈ M).
We define the following functions on T M : Fα(v) :=
ατ(v)(v, v) (v∈ T M), F := Fα+ ¯β, E := 1
2F 2.
Then F and E are smooth onT M .◦
Due to the non-degeneracy of α, there is a unique vector field βon M such that β(Y ) = α(β, Y ) for any vector field Y on M (Riesz’ lemma). Conversely, if X is a vector field on M , then we have a one-form X such that X(Y ) = α(X, Y ) for any vector field Y .
If β < 1, (M, E) is a Finsler manifold, called the Randers manifold obtained from the Riemannian manifold (M, α) by the perturbation with the one-form β.
Lemma 8.1 ([11]). Let (M, E) be the Randers manifold arising from the Rie-mannian manifold (M, α) by perturbation with β such that β < 1. Then the metric tensor g of (M, E) takes the form
g = F Fααˆ− ¯ β Fα3α¯⊗ ¯α + 1 Fαα¯ ˆβ + ˆβ ⊗ ˆβ,
where stands for the symmetric product.
In his paper [9], M. Matsumoto proved that β is a Killing vector field of the Randers manifold if and only if it is a Killing vector field of the original Riemannian manifold (M, α) as well. Now we use the results of section 5 to give a new proof of the sufficiency of this condition:
Proposition 8.2. Suppose that (M, α) is a Riemannian manifold, and X ∈ X(M) is a Killing vector field of (M, α) such that X < 1. Let β := X,
F := Fα+ ¯β and E = 12F2. Then X is a Killing vector field of the Randers
manifold (M, E).
Proof. First, suppose that X(p)= 0 at p ∈ M. Consider a chart (U, (ui)ni=1) around p and the induced chart (τ−1(U ), (xi)ni=1, (yi)ni=1) on T M . Let i, j ∈ {1, . . . , n} be arbitrary, then (LXg) ∂ ∂ui, ∂ ∂uj = Xcg ∂ ∂ui, ∂ ∂uj + g LX ∂ ∂ui, ∂ ∂uj + g ∂ ∂ui,LX ∂ ∂uj .
By the straightening-out theorem, we can choose a chart such that X = ∂u∂1. Then the last two terms vanish since, e.g.,
LX ∂ ∂ui = X, ∂ ∂ui = ∂ ∂u1, ∂ ∂ui = 0.
It remains to show that the first term also vanishes. We have the following coordinate expressions: ˆ α ∂ ∂ui, ∂ ∂uj = αvij, βˆ ∂ ∂ui = βiv, α¯ ∂ ∂ui = αvijyj, ¯ β = βivyi, Fα= αvijyiyj, F = αvijyiyj+ βivyi.
We substitute the expression in the preceding lemma for g: (LXg) ∂ ∂ui, ∂ ∂uj = ∂ ∂x1g ∂ ∂ui, ∂ ∂uj (∗) = ∂ ∂x1 1 + β v kyk (αvlmylym)1/2 αvij− β v kyk (αvlmylym)3/2α v irαvjsyrys + 1 (αvlmylym)1/2(α v iryrβjv+ βivαvjryr) + βviβjv , and βi = αijXj = αijδ1j = αi1.
On the other hand, since X is a Killing vector field of (M, α), we obtain 0 = L ∂ ∂u1α ∂ ∂ui, ∂ ∂uj = ∂αij ∂u1.
Thus we have shown that all functions in the square bracket of (∗) have van-ishing partial derivatives with respect to x1, and hence LXg = 0 on TpM if
X(p)= 0. On the other hand, if X(p) = 0, and there is a series (pn)∞n=0 such that pn→ p and X(pn)= 0 (n ∈ N), then LXg vanishes on TpM by continuity.
Finally, if there is a neighbourhood of p on which X vanishes, then LXg = 0
on TpM automatically.
8.2. Funk metrics
In this subsection we shall work on an open subset ofRn; Dv will denote the
directional derivative with respect to a vector v ∈ Rn and Di the ith partial
derivative (i = 1, . . . , n).
Let ϕ : Rn → R be a Minkowski functional [18], i.e., a function satisfying the following conditions:
(1) ϕ is continuous onRn and smooth onRn\ {0}; (2) ϕ(0) = 0, and ϕ(p) > 0 if p= 0;
(3) ϕ is positively homogeneous of degree 1;
(4) the second derivative ϕ(p) is non-degenerate (and thus necessarily pos-itive definite) if p= 0.
The set Ω := ϕ−1[0, 1[ is the interior of the indicatrix of ϕ. We shall use the canonical identification T Ω ∼= Ω×Rnand the natural projections π1: T Ω→ Ω
and π2 : T Ω→ Rn. A Finslerian fundamental function F : T Ω → R on Ω is determined by the relation
ϕ◦ π1+π2 F = 1 on T Ω.◦
The Finsler structure determined by F is traditionally called the Funk metrics on Ω. The Finsler energy is then E = 12F2. For more about Funk metrics, see [18].
Proposition 8.3. With notations and hypotheses as above, for a vector field X on Ω the following conditions are equivalent:
(1) X is a Killing vector field of (Ω, F );
(2) for every point p∈ Ω and vector v ∈ Rn such that p + v∈ ∂Ω, the vector X(p) + DvX(p) is parallel to the tangent hyperplane of ∂Ω in p + v.
Proof. Let (ui)ni=1 be the restriction of the canonical coordinate system of Rn to Ω and ((xi)n
i=1, (yi)ni=1) the induced coordinate system on T Ω. If the
coordinate expression of X is Xi ∂∂ui, its complete lift is
Xc =Xi v ∂ ∂xi + y j∂Xi ∂uj v ∂ ∂yi.
If we act by Xc on both sides of the relation defining F , we obtain
0 = Dkϕ◦ π1+ π2 F Xi v ∂ ∂xi xk+y k F +yj ∂Xi ∂uj v ∂ ∂yi xk+y k F = Dkϕ◦ π1+π2 F Xi v δki − y k F2 ∂F ∂xi +yj ∂Xi ∂uj v δki F − yk F2 ∂F ∂yi = Dkϕ◦ π1+π2 F Xk v +y j F ∂Xk ∂uj v −yk F2 Xi v ∂F ∂xi + y j∂Xi ∂ui v ∂F ∂yi = Dkϕ◦ π1+π2 F Xk v + y j F ∂Xk ∂uj v − yk F2X cF .
X is a Killing field if and only if XcF = 0. Furthermore, if vp(= 0) ∈ T Ω is arbitrary, and z := p +F (vv
p)(∈ ∂Ω), then
vkDkϕ(z) = gz(z, v)
ϕ(z) = gz(z, v)= 0. Therefore, it follows that X is a Killing field if and only if (∗) Dkϕ◦ π1+π2 F Xk v +y j F ∂Xk ∂uj v = 0.
From now on, we suppose that vp is of the form as in the proposition, i.e.,
p + v ∈ ∂Ω. By the homogeneity of F , if (∗) is satisfied for such vp’s, it is satisfied for all. In that case, F (vp) = 1, and evaluating (∗) at vp we obtain
(Dkϕ)(p + v)(Xk(p) + vjDjXk(p)) = (Dkϕ)(p + v)(Xk(p) + DvXk(p))
=gradϕ(p + v), X(p) + DvX(p) = 0,
or, equivalently, the vector X(p)+DvX(p) is parallel to the tangent hyperplane of the indicatrix at p + v.
§9. Discussion
It is known that a geodesic on a Riemannian manifold meets a translation at constant angles [2, 16, 26]. In the general case, if g is positive definite, the angle ϕ of a translation X and a geodesic c may be given by
cos ϕ(t) := g˙c(t)(X(c(t)), ˙c(t))
g˙c(t)(X(c(t)), X(c(t)))g˙c(t)( ˙c(t), ˙c(t)) .
The numerator is constant by 7.2, and the second factor in the denominator is constant as well even in the most general case. It follows from 7.3 that in the Riemannian case the first factor is also constant, since then the function g
ˆ X, ˆX
is constant on each fibre. From our results, however, it does not follow that the first factor is constant in general, even for Finsler manifolds. Therefore, it does not follow that ϕ is constant. It remains an open question whether there exists any class of metrics in which this angle is constant and which is more general than the Riemannian case.
Moreover, there is a broad class of metrics that have no non-trivial transla-tions at all. For example, the hyperbolic plane does not have any. In Poincar´e’s upper half-plane model with canonical coordintates (u1, u2) the Killing fields have the form
X = (αu1+ βu2+ γ) ∂ ∂u1 + αu
2 ∂
with some α, β, γ∈ R. If α = 0, the integral curves of X are given by c(t) = (c1+ βc2t)eαt−γ α, c2e αt,
with c1, c2 ∈ R, c2 > 0, which are no geodesics. That is, however, not sur-prising, since, if the hyperbolic plane had a non-trivial translation, a geodesic quadrangle with angle sum 2π could be constructed, in contradiction with the Gauss – Bonnet theorem.
In summary, we have tried to generalize some theorems of Riemannian geometry and Finsler geometry, and found that those not relying on the notion of translation may be successfully generalized.
Acknowledgement
I am grateful to my supervisor, Dr. J´ozsef Szilasi, for fruitful discussions and useful suggestions.
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Rezs˝o L. Lovas
Institute of Mathematics, University of Debrecen H – 4010 Debrecen, P.O. Box 12, Hungary
